Radial positive definite functions and Schoenberg matrices with negative eigenvalues

Abstract:

The main object under consideration is a class $\Phi _n\backslash \Phi _{n+1}$ of radial positive definite functions on $\mathbb {R}^n$ which do not admit radial positive definite continuation on $\mathbb {R}^{n+1}$. We find certain necessary and sufficient conditions on the Schoenberg representation measure $\nu _n$ of $f\in \Phi _n$ for $f\in \Phi _{n+k}$, $k\in \mathbb {N}$. We show that the class $\Phi _n\backslash \Phi _{n+k}$ is rich enough by giving a number of examples. In particular, we give a direct proof of $\Omega _n\in \Phi _n\backslash \Phi _{n+1}$, which avoids Schoenberg’s theorem; $\Omega _n$ is the Schoenberg kernel. We show that $\Omega _n(a\cdot )\Omega _n(b\cdot )\in \Phi _n\backslash \Phi _{n+1}$ for $a\not =b$. Moreover, for the square of this function we prove the surprisingly much stronger result $\Omega _n^2(a\cdot )\in \Phi _{2n-1}\backslash \Phi _{2n}$. We also show that any $f\in \Phi _n\backslash \Phi _{n+1}$, $n\ge 2$, has infinitely many negative squares. The latter means that for an arbitrary positive integer $N$ there is a finite Schoenberg matrix $\mathcal {S}_X(f) := \|f(|x_i-x_j|_{n+1})\|_{i,j=1}^{m}$, $X := \{x_j\}_{j=1}^m \subset \mathbb {R}^{n+1}$, which has at least $N$ negative eigenvalues.